Discrete Fourier transforms and Z-transforms are commonly used to analyze time domain signals and functions. A discrete Fourier transform transforms a function into a frequency domain representation of the original function, which is often a function in the time domain. Typically, a discrete Fourier transform requires an input function that is discrete and whose non-zero values have a limited (i.e., finite) duration. Inputs for discrete Fourier transforms are often created by sampling a continuous function (e.g., a person's voice). A Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
Time domain functions, discrete Fourier transforms and Z-transforms are related in the sense that one can be derived from any of the other. In other words, a discrete Fourier transform or a Z-transform can be derived from a time signal, a discrete Fourier transform or a time signal can be derived from a Z-transform, and a Z-transform or a time signal can be derived from a discrete Fourier transform.